The rule to arrive at (the unknown) capital with the aid of certain known and unknown profits (in a given transaction):-
222. By means of the operation of proportionate distribution, the (unknown) profits age to be determined from the mixed sum (of all the profits) minus the (known) profit. Then the capital of the person whose investment is unknown results from dividing his profit by that (same common factor which has been used in the process of proportionate distribution above).
An example in illustration thereof.
223-225. According to agreement some three merchants carried put (the operation of) buying and selling. The capital of the first (of them) consisted of six purāņas., that of the second of eight purāņas, but that of the third was not known. The profit obtained by all those (three) men was 96 purāņas. In fact the profit obtained by him (this third person) on the unknown capital happened to be 40 purāņas. What is the amount thrown by him (into the transaction), and what is the profit (of each) of the other two merchants? O friend, if you know the operation of proportionate distribution, tell (me this) after making the (necessary) calculation.
The rule for arriving at the wages (due in kind for having carried certain given things over a part of the stipulated distance according to a given rate):-
226.[*] From the square of the product (of the numerical value) of the weight to be carried and half of the (stipulated distance
226.^ Algebraically, the formula given in the rule is :
, where x=wages to be found out, a = the total weight to be carried, D= the total distance, d is the distance gone over, and b=the total wages promised. it may be noted here that the rate of the wages for the two stages of the journey is the same, although the amount paid for each stage of the journey is not in accordance with the promised rate for the whole journey.
The formula is easily derived from the following equation containing the data in the problem:-
